Wannier Topology and Quadrupole Moments for a generalized Benalcazar-Bernevig-Hughes Model

Abstract

We analyze a special separable and chiral-symmetric model with a quantized quadrupole moment, extending the Benalcazar-Bernevig-Hughes model [Science 357, 61 (2017)]. Using nested-Wilson loop formalism, we give an exact expression for Wannier centers, sector polarizations, and quadrupole moments. These are connected to the winding numbers of the constitutive one-dimensional chains. We prove that these winding numbers can characterize the model's Wannier topology as a $\mathbb{Z}\times \mathbb{Z}$ set. These results clearly show that the quantization of the quadrupole moment can arise without additional spatial symmetry (except for translation symmetry) for the bulk. By switching from the Wannier representation to the Bloch representation, we derive an alternative expression for the bulk quadrupole moment and obtain its exact value. Combining the bulk quadrupole and edge polarizations, we analytically calculate the corner charge in a large square system and make the bulk-boundary correspondence explicit in an edge-consistent gauge. Our work reveals the relationship between zero-energy states at the boundary, charge localization, and the bulk quadrupole of the extended model.

References in corpus (19)

• Topological Insulators with Inversion Symmetry
• Electric Multipole Moments, Topological Multipole Moment Pumping, and Chiral Hinge States in Crystalline Insulators
• $(d-2)$-dimensional edge states of rotation symmetry protected topological states
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Topological Defects and Gapless Modes in Insulators and Superconductors
• Novel Topological Phase with Zero Berry Curvature
• Acoustic realization of quadrupole topological insulators
• A General Theorem Relating the Bulk Topological Number to Edge States in Two-dimensional Insulators
• Engineering Corner States from Two-Dimensional Topological Insulators
• Experimental Measurement of the Quantum Metric Tensor and Related Topological Phase Transition with a Superconducting Qubit
• Topological Phase Transitions in Disordered Electric Quadrupole Insulators
• Chiral-Symmetric Higher-Order Topological Phases of Matter
• Exact Higher-order Bulk-boundary Correspondence of Corner-localized States
• Corner charge and bulk multipole moment in periodic systems
• Quadrupole moments, edge polarizations, and corner charges in the Wannier representation
• Thermodynamic approach to electric quadrupole moments
• Toeplitz operators on concave corners and topologically protected corner states
• Thermodynamic electric quadrupole moments of nematic phases from first-principles calculation
• Classification of topological invariants related to corner states

Cited by in corpus (3)

• Higher-order topological phases in crystalline and non-crystalline systems: a review
• Mixed higher-order topology and nodal and nodeless flat band topological phases in a superconducting multiorbital model
• Wilson loop invariants and bulk-boundary correspondence in higher-order topological insulators with two anticommuting mirror symmetries